Vectors and conics

Attempts to answer problems in areas as diverse as science, technology and economics...

Attempts to answer problems in areas as diverse as science, technology and economics involve solving simultaneous linear equations. In this unit we look at some of the equations that represent points, lines and planes in mathematics. We explore concepts such as Euclidean space, vectors, dot products and conics.

By the end of this unit you should be able to:

Section 1

recognise the equation of a line in the plane;

determine the point of intersection of two lines in the plane, if it exists;

recognise the one-one correspondence between the set of points in three-dimensional space and the set of ordered triples of real numbers;

recognise the equation of a plane in three dimensions.

Section 2

explain what are meant by a vector, a scalar multiple of a vector, and the sum and difference of two vectors;

represent vectors in and in terms of their components, and use components in vector arithmetic;

use the Section Formula for the position vector of a point dividing a line segment in a given ratio;

determine the equation of a line in or in terms of vectors.

Section 3

explain what is meant by the dot product of two vectors;

use the dot product to find the angle between two vectors and the projection of one vector onto another;

determine the equation of a plane in , given a point in the plane and the direction of a normal to the plane.

Section 4

explain the term conic section;

determine the equation of a circle, given its centre and radius, and the centre and radius of a circle, given its equation;

Vectors and conics

Introduction

The idea of vectors and conics may be new to you. In this unit we look at some of the ways that we represent points, lines and planes in mathematics.

In Section 1 we revise coordinate geometry in two-dimensional Euclidean space, 2, and then extend these ideas to three-dimensional Euclidean space, 3. We discuss the equation of a plane in 3, but find that we do not have the tools to determine the equation of a plane, and leave this until Section 3.

In Section 2 we introduce the idea of a vector, and look at the algebra of vectors. Vectors give us a way of looking at points and lines, in the plane and in 3, which is sometimes more useful than Cartesian coordinates, although the two are closely related.

In Section 3 we introduce the idea of the dot product of two vectors, and then use it to determine the general form of the equation of a plane in 3.

In Section 4 we explain the origin of conics, as the curves of intersection of double cones and planes in 3. The focus–directrix definitions of the non-degenerate conics, the ellipse, the parabola and the hyperbola, are given. We observe that conics are precisely the subsets of the plane determined by an equation of degree two.

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